Complex numbers – Grade 11 math lesson

Lesson 1 of 23 in the Grade 11 math course

US version. US grades, US spelling and dollars ($). · UK Maths Department

A complete 45-minute lesson for grade 11 (ages 16–17) that runs on this page – at school or at home. Every answer is graded right away, with a worked explanation.

Time: 45 minutes + 20 min homework
Grade: Grade 11 (ages 16–17)
You need: Paper, a pencil and this page
Graded: Instantly, with explanations

For parents: how to use this lesson

You don't need to know this topic yourself. The lesson explains each step on screen, one tidy step at a time. Every answer is graded straight away with a worked explanation, so you never have to grade anything.

Read what your child will learn and the big ideas together.
Press Start the lesson below and sit with your child for the teaching and the worked examples (about 19 minutes).
Let them do the practice on their own. They choose Getting started, On track or Challenge – start on On track and move up after a few right in a row.
Finish with the exit ticket and the talk question: “What happens to a complex number when you multiply it by i? Try it on the plane.”
Later that day or the next day: the homework (20 minutes), then a practice test.

Watch out for

Leaving i² in an answer. Replace i² with −1 straight away.
Adding a real part to an imaginary part. Keep them separate, like unlike terms in algebra.

Tip for home: Talk about why mathematicians invented numbers whose square is negative.

What your child will learn

Learning goal: add, subtract and multiply complex numbers and use i² = −1.

Success looks like

I can add and subtract complex numbers.
I can multiply complex numbers.
I can use the conjugate.

Key words

imaginary unit: i, where i² = −1
complex number: a number a + bi
real part: the a in a + bi
conjugate: a + bi and a − bi

What to have ready

Paper and a pencil
This page on a computer, tablet or phone
For the hands-on part: Plot complex numbers as points on a plane (real across, imaginary up) and see what adding does.

The big ideas (for the adult)

The number i is defined so that i² = −1. A complex number has the form a + bi, with a real part a and an imaginary part b.

Add and subtract complex numbers by combining the real parts and the imaginary parts separately.

Multiply like expanding parentheses, then replace i² with −1. The conjugate of a + bi is a − bi; their product is a real number.

The lesson, step by step

45 minutes in 5 parts
Part	Time	How it is checked
Starter: quick recall	5 min	Graded by Classfern
Teaching input	10 min	Not assessed, self-checked
Worked examples and guided practice	9 min	Not assessed, graded by Classfern
Independent practice	15 min	Graded by Classfern
Closure and exit ticket	6 min	Graded by Classfern, self-checked

Starter, "your turn", practice and exit ticket are graded by Classfern. Worked examples and teaching are not assessed. The hands-on task and the reflection are self-assessed. Explanations in talk are judged by the parent and not recorded.

The lesson

It runs here on the page. No sign-up and no names – progress is kept only in this browser.

Homework and practice at home

20 minutes: 10 practice questions, 3 multiple-choice questions and 2 to remember from earlier work, then a short reflection. New questions every time, all graded instantly.

More practice

All Grade 11 math lessons and tests · Math Department

Curriculum

Common Core State Standards HSN.CN.A.1: “Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.”

Common Core State Standards HSN.CN.B.5: “(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)³ = 8 because (-1 + √3 i) has modulus 2 and argument 120°.”

Our lessons and questions are our own. They are not written or endorsed by any government department, testing agency or exam board.